Consider two time-series variables, yt and xt. Generalizing the discussion about dynamic relationships in chapter 9 to these two interrelated variables yield a system of equations:
yt =віо + Piiyt-i + ві2Х*-і + (13.1)
xt =в20 + ^2iyt-1 + ^22Xt-1 + Vе (13.2)
The equations describe a system in which each variable is a function of its own lag, and the lag of the other variable in the system. Together the equations constitute a system known as a vector autoregression (VAR). In this example, since the maximum lag is of order one, we have a VAR(1).
If y and x are stationary, the system can be estimated using least squares applied to each equation. If y and x are not stationary in their levels, but stationary in differences (i. e., I(1)), then take the differences and estimate:
Ayt =6... Read More
In this chapter, you learned some basic concepts about probability. Since the actual values that economic variables take on are not actually known before they are observed, we say that they are random. Probability is the theory that helps us to express uncertainty about the possible values of these variables. Each time we observe the outcome of a random variable we obtain an observation. Once observed, its value is known and hence it is no longer random. So, there is a distinction to be made between variables whose values are not yet observed (random variables) and those whose values have been observed (observations). Keep in mind, though, an observation is merely one of many possible values that the variables can take. Another draw will usually result in a different value being observed.
The Garch-in-mean (MGARCH) model adds the equation’s variance to the regression function. This allows the average value of the dependent variable to depend on volatility of the underlying asset. In this way, more risk (volatility) can lead to higher average return. The equations are listed below:
yt = во + Oht + et (14.9)
ht = 5 + aief-1 + ydt-ie-! + eiht-i (14.10)
Notice that in this formulation we left the threshold term in the model. The errors are normally distributed with zero mean and variance ht.
The parameters of this model can be estimated using gretl, though the recursive nature of the likelihood function makes it a bit more difficult. In the script below (Figure 14.9) you will notice that we’ve defined a function to compute the log-likelihood.2 The function is called g... Read More
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B... Read More
The following script estimates the reduced form equations using least squares and the demand equation using two-stage least squares for Graddy’s Fulton Fish example.
In the example, ln(quan) and ln(price) are endogenously determined. There are several potential instruments that are available. The variable stormy may be useful in identifying the demand equation. In order for the demand equation to be identified, there must be at least one variable available that effectively influences the supply of fish without affecting its demand. Presumably, stormy weather affects the fishermen’s catch without affecting people’s appetite for fish! Logically, stormy may be a good instrument.
The model of demand includes a set of indicator variables for day of the week... Read More
Obtaining confidence intervals for the marginal effects (and the AME) is relatively straightforward as well. To estimate the standard error of the marginal effect, we resort to the Delta method. This method of finding the variance of functions of parameters was discussed in section 5.3.2. You may want to take a look at this section again (page 99), before proceeding.
Using the Delta method means taking analytic or numerical derivatives of the marginal effect or AME to be used in the computation of the standard error of the AME. The analytic derivatives are not that hard to take, but why bother when numerical ones are available. This is the approach taken in commercial software that includes the ability to estimate nonlinear combinations of parameters and their standard errors.
The functi... Read More